_The subject of Rotatory Motion continued.--A Ball, by having a
peculiar spinning motion imparted to it, may be made to stop
short, or to retrograde, though it meets not with any apparent
obstacle.--The rectilinear path of a Spherical Body influenced
by its rotatory motion.--Bilboquet, or Cup and Ball.--The joint
forces which enable the Balancer to throw up and catch his Balls
on the full gallop.--The Hoop.--The Centre of Percussion.--The
Whip and Peg-top.--Historical Notices.--The power by which the
Top is enabled to sustain its vertical position during the act
of spinning.--The sleeping of the Top explained.--The force
which enables it to rise from an oblique into a vertical
position.--Its gyration._
“Tom, do you remember that I told you a few days ago,” said Mr.
Seymour, “that, by giving a revolving body a peculiar _spinning_
motion, certain effects were produced, which I should, on some future
occasion, take into consideration?”
“To be sure I do,” replied Tom.
“Well, then, attend to me.”
Mr. Seymour took a marble, and, placing it on the ground, gave it an
impulse forward by pressing his forefinger upon it: the marble darted
forward a few paces, after which it rolled back again.
“That is most extraordinary!” cried Tom; “the marble came back to your
hand, as it were, of its own accord, and without having met with any
obstacle.”
“And you, no doubt,” said Mr. Seymour, “regard it as contrary to the
well known law, that a body once put in motion, in any direction, will
continue to move in that direction until some foreign cause oppose it.”
“It really would appear so.”
“It is, however, far otherwise; the force which I imparted to the
marble communicated to it two kinds of motion; the one projecting it
forward, the other producing a rotatory motion round its axis, in a
direction opposite to that of its rectilinear course; and the
consequence was simply this, that when the former motion, on account of
the friction of the marble on the ground, was destroyed, the rotatory
motion continued, and by thus establishing an action in an opposite
direction, caused the marble to retrograde. If, however, you will fetch
your hoop, I will demonstrate the fact on a larger scale.”
Tom accordingly produced the hoop; and Mr. Seymour projected it
forward, giving to it, at the same instant, a spinning motion in an
opposite direction. The hoop proceeded forward to a certain distance,
when it stopped, and then ran back to the hand.
“Let me beg you,” said Mr. Seymour, “to treasure this fact in your
memory; you perceive by it how greatly the progressive direction of a
body may be influenced by a rotatory motion around its axis; and,
indeed, the theory of the _rifle_ gun(20) is easily deduced from it. It
will also explain the effect which a rotatory motion produces in
steadying or disturbing the direction of a projectile. It is for such a
reason that the balancer constantly whirls round his balls or oranges,
as he throws them into the air, with the intention of catching them
again; and that in playing at _Bilboquet_, or cup and ball, you find it
necessary to give a spinning motion to the ball, in order to catch it
on the spike--but we will consider this subject presently. I am now
desirous of laying down a few propositions upon the subject of
rotation, the knowledge of which is essential for the explanation of
the motions of revolving bodies.”
Mr. Seymour proceeded to state that every body had three principal axes
upon which it might revolve, but that the shortest was the only one
upon which it could permanently and steadily rotate--that should it, in
consequence of the impulse given to it, begin to spin upon any other
than the shortest axis, it would, during its revolutions, be constantly
showing a tendency to approach it; whence it followed that, under such
circumstances, it would be unsteady and _wabbling_ in its motions.
In order, however, to make this proposition intelligible to the
children, Mr. Seymour performed the following simple experiment.
[Illustration: Two rings, labeled 1 and 2.]
Having tied some string to a common curtain ring, as represented by
figure 1, he twisted it round by means of his thumb and finger, until
it acquired considerable velocity, when the ring was seen to rise
gradually into the position represented by fig. 2. Thus, in the
simplest manner, was a revolving body shown to exchange its longer for
its shorter axis.
The children declared that they perfectly comprehended the subject, and
Tom observed that, in future, whenever he wished to make a ball spin
steadily, he should take care to make it turn on its shortest axis.
“You are quite right, Tom,” said Mr. Seymour; “and the skilful bowler
at cricket, in order to give his ball a steady axis of rotation, always
holds it with the seam across, so that the tips of his fingers may
touch, and he takes care to hold it only with such a grasp as may be
sufficient to steady it, for by a turn even of the wrist it may be made
to proceed unsteadily; and this leads me to consider another equally
important proposition--viz. that the axis of rotation should coincide
with the direction in which it is moving forward, or, in other words,
with its line of motion. Now, where this is not the case, it is evident
that the unequal action of the air will cause the body to deviate from
its straight course, since its two sides, having different velocities
(the rotatory and progressive motions conspiring on one side, while
they are in opposition on the other), will be differently affected by
such resistance; the resistance, of course, increasing with the
velocity. It is upon this principle,” continued Mr. Seymour, “that Sir
Isaac Newton has explained the irregular motion of the tennis-ball.”
“But do explain to us, papa,” said Louisa, “why it is so necessary to
spin the ball in order to catch it on the spike?”
“Rotatory motion, my dear, when directed according to the principles I
have endeavoured to enforce, will always steady the course of a body.
In playing at bilboquet, your object is so to throw up the ball that
its hole may descend perpendicularly upon the spike which is held for
its reception; and in order to accomplish this, you make the ball spin
upon an axis, at the extremity of which is the hole; the consequence is
obvious.”
Louisa observed, that she well remembered an allusion to this game in
Miss Edgeworth’s Essays on Education; and that, unless she was much
deceived, the advantage to be gained by spinning the ball was referred
to centrifugal force, and its effect in preserving the “_parallelism of
motion_.”
“I do not recollect the passage,” answered her father, “but I will
admit that the centrifugal force is indirectly instrumental to the
effect, although, in my view of the subject, it is more philosophical
to refer it at once to the creation of an appropriate axis of rotation.”
“I well remember,” observed Tom, “that the rider at Astley’s whirled
round the oranges as he threw them into the air.”
“And I hope that you are now not only acquainted with the principle
which rendered such a rotatory motion necessary, but that which must
make the shorter the more eligible axis for effecting his purpose;--but
can you tell me how it could have happened, that the oranges, which
were thrown perpendicularly upwards while the horseman was on the full
gallop, should have fallen again into his hand?”
“Ay,” said Louisa, “that puzzled me exceedingly; I should have thought
he would have ridden away from them, and that they must have fallen
several feet behind him.”
“What say you, Tom, to that?” enquired Mr. Seymour.
“I suppose,” replied Tom, “that the rider calculated upon the distance
he would pass forward before they could fall, and projected them
accordingly.”
“No, indeed; there is no calculation in the case, nor is any art used
to throw the oranges in advance: they are projected perpendicularly
from the hand; and if you will only recall to your mind the subject of
the ‘Composition of Forces,’ the mystery will vanish.”
“I see it all clearly,” cried Tom; “the orange partakes of the
progressive motion of the rider; when, therefore, he throws it upwards,
it is influenced by two forces which are in the direction of the two
sides of a parallelogram, and it consequently describes the diagonal.”
“You are quite right; but you doubtless will perceive that, instead of
a straight line, the orange will describe a parabolic curve.”
“For the same reason, I suppose,” said Tom, “that the stone from the
sling described a curve?”
“Certainly; but see, I have a diagram which will explain the subject
more clearly.”
[Illustration: A man on a moving horse demonstrates progressive motion
with an orange.]
“The orange, as it is thrown into the air, is influenced by two forces:
the one arising from the progressive motion of the rider, the other
from the projectile force imparted to it. These two forces are in the
direction of the adjacent sides of a parallelogram, and were it not for
the operation of gravity, the body would accordingly describe its
diagonal in the same space of time as it would have described one of
the sides.[18] The influence of gravity, however, not only deflects it
from a right line into a curve, but diminishes its force, so that
instead of arriving at the opposite angle of the parallelogram _a_, its
greatest altitude will be short of that point; it will then descend
through a similar curve; and, since the times of ascent and descent are
equal,[19] it will reach the hand of the rider at the very moment he is
prepared to receive it; for the orange will have traversed the
parabolic curve in the same space of time as the horseman required for
passing from one extremity of the curve to the other.”
Mr. Seymour having concluded this explanation, much to the satisfaction
of the young party, observed that the present occasion was an
appropriate one for the introduction of some remarks on the favourite
pastime of the HOOP.
“It is a classical pastime,” exclaimed the vicar, “and was as common
with the Greeks and Romans as it is with boys of the present
generation.”
“And it has the advantage,” added Mr. Seymour, “of sending the tide of
life in healthful currents through the veins.”
Tom began to trundle his hoop along the gravel walk.
“Stop, stop, my dear boy,” cried his father, “you seem to have
forgotten our compact, that every toy should be fairly won before it
was played with. Come upon the lawn, and let me ask you some questions
relative to the motions of the hoop. Can you make it stand still upon
its edge?”
“Not readily,” was Tom’s reply.
“And yet,” continued Mr. Seymour, “during its progressive motion, it
rolls on its edge without any disposition to fall: how happens that?”
“It is owing to the centrifugal force, which gives it a motion in the
direction of a _tangent_ to the circle, and, consequently, overcomes
the force of gravity.”
“Your answer is pat,” replied his father: “as long as you give your
hoop a certain degree of velocity, the _tangential_, or centrifugal
force, overcomes gravity, in the manner you have already witnessed;[20]
but, when that is slackened, the hoop will fall on its side; not,
however, until it has made several complete revolutions. Now, answer me
another question. Why is it so difficult to make the hoop proceed
straight forward, without turning to the right or left?”
“I suppose it arises from the same cause as that which altered the
direction of my marble as it ran along--the inequality of the ground.”
“That,” replied his father, “would undoubtedly have its influence; but
it is principally to be referred to the impossibility of your
constantly giving a straight blow by the stick. When it is moving
forward, a slight inclination towards either side will cause the parts
to acquire a motion towards that side, those which are uppermost being
most affected by it; and this lateral, or sideway motion, assisted
sometimes by the irregular curvature of the hoop, causes its path to
deviate from a rectilinear direction; so that, instead of moving
straight forward, it turns to that side towards which it began to
incline; and, in this position, its tendency to fall is still farther
counteracted by the centrifugal force. It is from a similar cause that
the bullet, unless rifled, will have a tendency to go to the right or
left, from any unequal impulse which it may have received at the moment
of its exit from the barrel. I have yet one other question, and, as its
answer will lead us into the consideration of a mechanical subject of
some importance, I must beg you to bestow all your attention. In
trundling your hoop, have you not often observed that, although the
blow inflicted upon it by your stick might have been violent, yet the
effect produced by it was comparatively small, in consequence of the
hoop having been struck by a disadvantageous part of the stick?”
“Certainly! I have frequently observed that, if the hoop is struck by
the stick either too near the hand, or too near the end, much of its
force is lost; and I have also noticed the same thing in striking the
ball with my cricket-bat.”
“The fact is,” said Mr. Seymour, “that every striking body has what is
termed its _centre of percussion_, in which all the percutient force of
a body is, as it were, collected; thus, a stick of a cylindrical
figure, supposing the centre of motion at the hand, will strike the
greatest blow at a point about two-thirds of its length from the wrist.
I may, perhaps, at some future time, return to this subject, and
explain several mechanical effects which are dependent upon it.(21) Now
away with you, and trundle your hoop, or spin your top; as soon as the
vicar arrives I will rejoin you.”
In the course of an hour Mr. Seymour and his reverend friend proceeded
to the play-ground, where they found the children busily engaged in
their several diversions.
“I rejoice to find you at so classical a pastime,” said the vicar, as
he approached Tom, who was busily engaged in spinning his top. “The
top, my boy, is a subject which the great Mantuan bard did not consider
beneath the patronage of his muse: but, hey-day! this is not the
‘_volitans sub verbere turbo_’ of the immortal Virgil; the top of
antiquity was the whip-top, the peg-top is a barbarous innovation of
modern times: a practical proof of the degeneracy of the race. Even
boys, forsooth, must now-a-days have their activity cramped by
inventions to supersede labour: well may we regard the weapons, which
our sturdy ancestors wielded as instruments rather calculated for
giants than men, if such pains be taken to instil into the minds of
youth the mischievous spirit of idleness.”
“My dear sir,” said Tom, who was always grieved at displeasing the
vicar, “if it will gratify you, I will spin my _whip_-top, for I have
an excellent one which my papa has lately given me.”
“Well said! my dear boy. ‘_Puer bonæ spei._’--What a pity would it be
to damp so noble a spirit; get your whip-top.”
Tom accordingly placed the Virgilian top upon the ground, and as the
boy plied the whip, so did the vicar lash the air with his quotation;
running round the top in apparent ecstasy, while he repeated the
well-known lines from the seventh Æneid:--
“Ille actus habena
Curvatis fertur spatiis; stupet inscia turba,
Impubesque manus, mirata volubile buxum:
Dant animos plagæ.”[21]
As Mr. Twaddleton thus gave vent to that fervour which was ever kindled
by collision with Virgil, Tom gave motion to his top, which swaggered
about with such an air of self-importance, that, to the eye of fancy,
it might have appeared as if proudly conscious of the encomiums that
had been so liberally lavished upon it.
“The Grecian boys, as Suidas informs us, played also with this top,”
continued the vicar.
“And pray, may I ask,” said Mr. Seymour, “whether it was not introduced
into this country by the Romans?”
“Probably,” replied the vicar. “Figures representing boys in the act of
whipping their tops first appear in the marginal paintings of the
manuscripts of the fourteenth century; at which period, the form of the
toy was the same as it is at present, and the manner of impelling it by
the whip can admit of but little if any difference. In a
manuscript,[22] at the British Museum, I have read a very curious
anecdote which refers to Prince Henry, the eldest son of James the
First; with your permission I will relate it to you.”
Here the vicar extracted a memorandum-book from his pocket, and read
the following note:--
“The first tyme that he, the prince, went to the towne of Sterling to
meete the king, seeing a little without the gate of the towne a stack
of corne, in proportion not unlike to a topp, wherewith he used to
play, he said to some that were with him, ‘Loe there is a goodly topp:’
whereupon one of them saying, ‘Why doe you not play with it then?’ he
answered, ‘Set you it up for me, and I will play with it.’”
“Was not that a clever retort of the young prince?” said the vicar, as
he returned the manuscript into his memorandum-book; “and I think it
must have confounded the courtier who could have asked so silly a
question.”
“Well, Tom,” said Mr. Seymour, “let us see whether you can set up your
own top, so that it shall stand steadily on its point.”
“I have often tried that experiment,” answered Tom, “but could never
succeed in keeping the line of direction within its narrow base.”
“And yet, when in rotatory motion, its erect position is maintained
without difficulty; how is that?”
“Is it not owing to the centrifugal force?” asked Tom.
“Undoubtedly: but as the subject is highly interesting, I will
endeavour to explain it more fully. You must, however, first obtain
permission from the vicar to spin your humming-top, for that will
better illustrate the phenomena which it is my wish to examine.”
“If your object is the exercise of the body, let us spin the whip-top,”
replied the vicar; “but if you wish to exercise the boy’s mind, I
cannot object to your selecting the top best calculated to fulfil that
desire.”
Tom, having accordingly prepared his top, pulled the string, and set
the wooden machine spinning on the floor.
“Now, Tom, I will explain to you the reason of the top being able to
sustain its vertical position. You have already learned, from the
action of the sling, that a body cannot move in a circular path,
without making an effort to fly off in a right line from the
centre;[23] so that, if a body be affixed to a string, and whirled
round by the hand, it will stretch it, and in a greater degree
according as the circular motion is more rapid.”
“Certainly,” said Tom.
“The top, then, being in motion, all its parts tend to recede from the
axis, and with greater force the more rapidly it revolves: hence it
follows that these parts are like so many powers acting in a direction
perpendicular to the axis; but, as they are all equal, and as they pass
all round with rapidity by the rotation, the result must be that the
top is in equilibrio on its point of support, or on the extremity of
the axis on which it turns. But see, your top is down.”
“And what is the reason,” asked Tom, “of its motion being stopped?”
“I can answer that question, papa,” said Louisa; “is it not owing to
the friction of the ground?”
“Certainly; that has, doubtless, its influence: but the resistance of
the air is also a powerful force upon this occasion. A top has been
made to spin in vacuo as long as two hours and sixteen minutes.[24] But
come, Tom, spin your top once more. Observe,” exclaimed Mr. Seymour,
“how obliquely the top is spinning. It is now gradually rising out of
an oblique position;--now it is steadily spinning on a vertical
axis;--and now its motion is so steady, that it scarcely seems to move.”
“It is _sleeping_, as we call it,” said Tom.
“Its centre of gravity is now situated perpendicularly over its point
of support, which is the extremity of the axis of rotation: but attend
to me,” continued Mr. Seymour, “for I am about to attempt the
explanation of a phenomenon which has puzzled many older and wiser
philosophers than yourselves. It is evident that the top, in rising
from an oblique to a vertical position, must have its centre of gravity
raised; what can have been the force which effected this change?”
“Was it the centrifugal force?” asked Tom.
“Certainly not,” said Mr. Seymour, “as I will presently convince you.”
“Then it must have been the resistance of the air,” said Louisa.
“No; nor was it the resistance of the air,” replied her father: “for
the same effect takes place in vacuo.”
“Then pray inform us, by what means the top was raised.”
[Illustration: A toy top, diagonal to a surface.]
“It entirely depended upon the form of the extremity of the peg, and
not upon any simple effect connected with the rotatory or centrifugal
force of the top. I will first satisfy you that, were the peg to
terminate in a fine, that is to say, in a _mathematical_ point, the top
never could raise itself. Let A B C be a top spinning in an oblique
position, having the end of the peg, on which it spins, brought to a
fine point. It will continue to spin in the direction in which it
reaches the ground, without the least tendency to rise into a more
vertical position; and it is by its rotatory or centrifugal force that
it is kept in this original position: for if we conceive the top
divided into two equal parts A and B, by a plane passing through the
line X C, and suppose that at any moment during its spinning, the
connection between these two parts were suddenly dissolved, then would
any point in the part A fly off with the given force in the direction
of the tangent, and any corresponding point in the part B with an equal
force in an opposite direction; whilst, therefore, these two parts
remain connected together, during the spinning of the top, these two
equal and opposite forces A and B will balance each other, and the top
will continue to spin on its original axis. Having thus shown that the
rotatory or centrifugal force can never make the top rise from an
oblique to a vertical position, I shall proceed to explain the true
cause of this change, and I trust you will be satisfied that it depends
upon the bluntness of the point. Let A B C be a top spinning in an
oblique position, terminating in a very short point with a
hemispherical shoulder P _a_ M. It is evident that, in this case, the
top will not spin upon _a_ the end of the true axis X _a_, but upon P,
a point in the circle P M, to which the floor I F is a tangent.
Instead, therefore, of revolving upon a fixed and stationary point, the
top will roll round upon the small circle P M on its blunt point, with
very considerable friction, the force of which may be represented by a
line O P at right angles to the floor I F, and to the spherical end of
the peg of the top: now it is the action of this force, by its pressure
on one side of the blunt point of the top, which causes it to rise in a
vertical direction. Produce the line O P till it meets the axis C; from
the point C draw the line C T perpendicular to the axis _a_ X, and T O
parallel to it; and then, by a resolution of forces, the line T C will
represent that part of the friction which presses at right angles to
the axis, so as gradually to raise it in a vertical position; in which
operation the circle P M gradually diminishes by the approach of the
point P to _a_, as the axis becomes more perpendicular, and vanishes
when the point P coincides with the point _a_, that is to say, when the
top has arrived at its vertical position, where it will continue to
_sleep_, without much friction, or any other disturbing force, until
its rotatory motion fails, and its side is brought to the earth by the
force of gravity.”
[Illustration: A top diagonal to a surface, with a diagram describing
its rotation.]
“I _think_ I understand it,” said Tom, “although I have some doubt
about it; but if you would be so kind as to give me the demonstration
in writing, I will diligently study it.”
“Most readily,” said Mr. Seymour. “Indeed I cannot expect that you
should comprehend so difficult a subject, without the most patient
investigation; and, in the present state of your knowledge, I am
compelled to omit the relation of several very important circumstances,
to which I may, hereafter, direct your attention. When, for instance,
you have become acquainted with the elements of astronomy, I shall be
able to show you that the gyration of the top depends upon the same
principles as the precession of the equinoxes.”(22)
[Illustration: Two boys spin a top with whips.]
-----
Footnote 18:
See page 119.
Footnote 19:
See page 54.
Footnote 20:
See page 126.
Footnote 21:
“The wooden engine flies and whirls about,
Admired, with clamours, of the beardless rout:
They lash aloud; each other they provoke,
And lend their little souls at every stroke.”--DRYDEN.
Footnote 22:
Harl. lib. i. marked 6391.
Footnote 23:
Page 126.
Footnote 24:
Short on ‘Serson’s Horizontal Top.’ Phil. Trans. xlvii. 352.
------------------------------------------------------------------------
[Illustration: A boy playing on a swing between two trees.]